spin-warp pulse sequence rf g - cfmriweb.ucsd.edu1! tt. liu, be280a, ucsd fall 2012! bioengineering...

1

TT. Liu, BE280A, UCSD Fall 2012

Bioengineering 280A ��Principles of Biomedical Imaging��

��Fall Quarter 2013��

MRI Lecture 3��


Spin-Warp Pulse Sequence

Gx(t)

Gy(t)

RF

ky Gy(t)


Spin-Warp Gx(t)

t1 ky

Gy(t)

kx


2

TT. Liu, BE280A, UCSD Fall 2012 Thomas Liu, BE280A, UCSD, Fall 2008

Sampling in k-space


Aliasing


Aliasing

∆Bz(x)=Gxx Faster Slower TT. Liu, BE280A, UCSD Fall 2012

Intuitive view of Aliasing

€

kx =1/FOV

FOV

€

kx = 2 /FOV

3


Fourier Sampling

Instead of sampling the signal, we sample its Fourier Transform

???

Sample

F-1

F


Fourier Sampling

€

GS (kx ) =G(kx )1Δkx

comb kxΔkx

#

$ %

&

' (

=G(kx ) δ(n=−∞

∞

∑ kx − nΔkx )

= G(nΔkx )δ(n=−∞

∞

∑ kx − nΔkx )

kx

(1 /Δkx) comb(kx /Δkx)

Δkx


Fourier Sampling -- Inverse Transform

Χ =

*

1/Δkx

Δkx

=


Nyquist Condition

FOV (Field of View)

1/Δkx

To avoid overlap, 1/Δkx> FOV, or equivalently, Δkx<1/FOV

4


Aliasing

FOV (Field of View)

1/Δkx

Aliasing occurs when 1/Δkx< FOV


Aliasing Example

Δkx=1

1/Δkx=1


2D Comb Function

€

comb(x,y) = δ(x −m,y − n)n=−∞

∞

∑m=−∞

∞

∑

= δ(x −m)δ(y − n)n=−∞

∞

∑m=−∞

∞

∑= comb(x)comb(y)


Scaled 2D Comb Function

€

comb(x /Δx,y /Δy) = comb(x /Δx)comb(y /Δy)

= ΔxΔy δ(x −mΔx)δ(y − nΔy)n=−∞

∞

∑m=−∞

∞

∑

Δx

Δy

5


X =

* =

1/∆k


2D k-space sampling

€

GS (kx,ky ) =G(kx,ky )1

ΔkxΔkycomb kx

Δkx,kyΔky

#

$ % %

&

' ( (

=G(kx,ky ) δ(n=−∞

∞

∑ kx −mΔkx,ky − nΔky )m=−∞

∞

∑

= G(mΔkx,nΔky )δ(n=−∞

∞

∑ kx −mΔkx,ky − nΔky )m=−∞

∞

∑


Nyquist Conditions

FOVX

FOVY

1/∆kX

1/∆kY

1/∆kY> FOVY

1/∆kX> FOVX


Windowing

€

GW kx,ky( ) =G kx,ky( )W kx,ky( )

€

gW x,y( ) = g x,y( )∗w(x,y)

Windowing the data in Fourier space

Results in convolution of the object with the inverse transform of the window

6


Resolution


Windowing Example

€

W kx,ky( ) = rect kxWkx

"

# $ $

%

& ' ' rect

kyWky

"

# $ $

%

& ' '

€

w x,y( ) = F −1 rect kxWkx

#

$ % %

&

' ( ( rect

kyWky

#

$ % %

&

' ( (

)

* + +

,

- . .

=WkxWky

sinc Wkxx( )sinc Wky

y( )

€

gW x,y( ) = g(x,y)∗∗WkxWky


y( )


Effective Width

€

wE =1

w(0)w(x)dx

−∞

∞

∫

wE

€

wE =11

sinc(Wkxx)dx

−∞

∞

∫

= F sinc(Wkxx)[ ]

kx = 0

=1Wkx

rect kxWkx

%

& ' '

(

) * * kx = 0

=1Wkx

Example

€

1Wkx

€

−1Wkx


Resolution and spatial frequency

€

2Wkx

€

With a window of width Wkx the highest spatial frequency is Wkx

/2.This corresponds to a spatial period of 2/Wkx

.

€

1Wkx

= Effective Width

7


Windowing Example

€

g(x,y) = δ(x) + δ(x −1)[ ]δ(y)

€

gW x,y( ) = δ(x) + δ(x −1)[ ]δ(y)∗∗WkxWky


y( )=Wkx

Wkyδ(x) + δ(x −1)[ ]∗ sinc Wkx

x( )( )sinc Wkyy( )

=WkxWky

sinc Wkxx( ) + sinc Wkx

(x −1)( )( )sinc Wkyy( )

€

Wkx=1

€

Wkx= 2

€

Wkx=1.5


Sampling and Windowing

X =

* =

X

* =


Sampling and Windowing

€

GSW kx,ky( ) =G kx,ky( ) 1ΔkxΔky

comb kxΔkx

,kyΔky

#

$ % %

&

' ( ( rect

kxWkx

,kyWky

#

$ % %

&

' ( (

€

gSW x,y( ) =WkxWkyg x,y( )∗∗comb(Δkxx,Δkyy)∗∗sinc(Wkx

x)sinc(Wkyy)

Sampling and windowing the data in Fourier space

Results in replication and convolution in object space.


Sampling in ky

kx

ky

Gx(t)

Gy(t)

RF

Δky

τy

Gyi

€

Δky =γ2π

Gyiτ y

€

FOVy =1Δky

8


Sampling in kx

x Low pass Filter ADC

x Low pass Filter ADC €

cosω0t

€

sinω0t

RF Signal

One I,Q sample every Δt

M= I+jQ

I

Q

Note: In practice, there are number of ways of implementing this processing.


Sampling in kx

kx

ky

€

Δkx =γ2π

GxrΔt

€

FOVx =1Δkx

Gx(t)

t1

ADC

Gxr

Δt


Resolution

€

δx =1Wkx

= 12kx,max

= 1γ

2πGxrτ x

€

Wkx€

Wky

Gx(t) Gxr

€

τ x

€

δy =1Wky

= 12ky,max

= 1γ

2π2Gypτ y

Gy(t)

τy €

Gyp


Example

€

Goal :FOVx = FOVy = 25.6 cmδx = δy = 0.1 cm

€

Readout Gradient :

FOVx =1

γ2π

GxrΔt

Pick Δt = 32 µsec

Gxr =1

FOVxγ

2πΔt

=1

25.6cm( ) 42.57 ×106T−1s−1( ) 32 ×10−6 s( )

= 2.8675 ×10−5 T/cm = .28675 G/cm

1 Gauss = 1×10−4 Tesla

t1

ADC

Gxr

Δt

9


Example

€

Readout Gradient :

δx =1

γ2π

Gxrτ x

τ x =1

δxγ

2πGxr

=1

0.1cm( ) 4257 G−1s−1( ) 0.28675 G/cm( )

= 8.192 ms = NreadΔtwhere

Nread =FOVxδx

= 256

Gx(t)

Gxr

€

τ x


Example

€

Phase - Encode Gradient :

FOVy =1

γ2π

Gyiτ y

Pick τ y = 4.096 msec

Gyi =1

FOVyγ

2πτ y

=1

25.6cm( ) 42.57 ×106T−1s−1( ) 4.096 ×10−3 s( )

= 2.2402 ×10-7 T/cm = .00224 G/cm

τy

Gyi


Example

€

Phase - Encode Gradient :δy =

1γ

2π2Gypτ y

Gyp =1

δy 2 γ2π

τ y

=1

0.1cm( ) 4257 G−1s−1( ) 4.096 ×10-3 s( ) = 0.2868 G/cm

=Np

2Gyi

where

Np =FOVyδy

= 256

Gy(t)

τy €

Gyp


Sampling

€

Wkx€

Wky

In practice, an even number (typically power of 2) sample is usually taken in each direction to take advantage of the Fast Fourier Transform (FFT) for reconstruction.

ky

y

FOV/4

1/FOV

4/FOV

FOV

10


Example Consider the k-space trajectory shown below. ADC samples are acquired at the points shownwith

€

Δt = 10 µsec. The desired FOV (both x and y) is 10 cm and the desired resolution (both xand y) is 2.5 cm. Draw the gradient waveforms required to achieve the k-space trajectory. Labelthe waveform with the gradient amplitudes required to achieve the desired FOV and resolution.Also, make sure to label the time axis correctly.

PollEv.com/be280a TT. Liu, BE280A, UCSD Fall 2012 GE Medical Systems 2003


Gibbs Artifact

256x256 image 256x128 image

Images from http://www.mritutor.org/mritutor/gibbs.htm

* =


Apodization

Images from http://www.mritutor.org/mritutor/gibbs.htm

* =

rect(kx)

h(kx )=1/2(1+cos(2πkx) Hanning Window

sinc(x)

0.5sinc(x)+0.25sinc(x-1) +0.25sinc(x+1)

11


Aliasing and Bandwidth

x

LPF ADC

ADC €

cosω0t

€

sinω0t

RF Signal

I

Q LPF

x

*

x f

t

x

t

FOV 2FOV/3

Temporal filtering in the readout direction limits the readout FOV. So there should never be aliasing in the readout direction.


Aliasing and Bandwidth

Slower

Faster x

f

Lowpass filter in the readout direction to prevent aliasing.

readout

FOVx

B=γGxrFOVx

TT. Liu, BE280A, UCSD Fall 2012 GE Medical Systems 2003

spin-warp pulse sequence rf g - cfmriweb.ucsd.edu1! tt. liu, be280a, ucsd fall 2012! bioengineering...

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